The isoperimetric inequality for the Ky Fan norm

Technical Summary: The Isoperimetric Inequality for the Ky Fan Norm

Problem Statement
The paper addresses a conjecture by Nicola, Riccardi, and Tilli regarding the isoperimetric properties of Toeplitz operators on the Bargmann–Fock space $F^2(\mathbb{C})$. Specifically, it investigates the maximization of the sum of the first $N$ eigenvalues of the Toeplitz operator $T_\Omega$ with symbol $1_\Omega$ (the indicator function of a measurable set $\Omega \subset \mathbb{C}$ of finite area $s$).

While it was previously established that a disc maximizes the largest eigenvalue $\lambda_1(\Omega)$ (the Faber–Krahn inequality for the short-time Fourier transform), the behavior of sums of eigenvalues $\sum_{k=1}^N \lambda_k(\Omega)$ for $N > 1$ is more complex. For individual eigenvalues with $k > 1$, the isoperimetric inequality fails; for instance, among radial sets, an annulus maximizes $\lambda_2(\Omega)$, and circular symmetry breaks down entirely if the radial assumption is removed. The central question is whether the "fine averaging" of energy inherent in the Ky Fan norm (the sum of the first $N$ eigenvalues) restores the optimality of the disc for arbitrary measurable sets, without requiring radial symmetry.

Methodology
The proof employs a novel synthesis of spectral theory, quantum information theory, and rearrangement inequalities. The methodology proceeds through three main steps:

1. Ky Fan Representation via Density Operators:
   The author utilizes Ky Fan's maximum principle to reframe the sum of the first $N$ eigenvalues. Instead of viewing the sum as a property of the operator $T_\Omega$ acting on individual functions, it is represented as the maximum concentration of the Husimi function associated with rank-$N$ orthogonal projections. Specifically, for a density operator $\rho_P = \frac{1}{N}P$ where $P$ is a rank-$N$ projection, the sum is given by:
   $$\frac{1}{N} \sum_{j=1}^N \lambda_j(\Omega) = \max_{P} \int_\Omega Q_{\rho_P}(z) \, dA(z)$$
   where $Q_{\rho_P}$ is the Husimi function of the state $\rho_P$.

2. De Palma's Majorization Theorem:
   The argument leverages a deep result from quantum information theory by De Palma, Trevisan, and Giovannetti. This theorem states that among all density operators with a fixed spectrum, the "passive rearrangement" (where eigenvalues are ordered decreasingly along the Fock basis) maximizes the concentration of the Husimi function in the sense of convex integrals. By applying this to the specific case of rank-$N$ projections, the author shows that the optimal state for maximizing the integral over any set $\Omega$ is the one where the eigenvalues are distributed among the lowest energy Fock states ($e_0, \dots, e_{N-1}$).

3. The Bathtub Principle:
   Once the optimal state is identified as the one diagonal in the Fock basis with decreasing eigenvalues, its Husimi function is shown to be radial and strictly decreasing away from the origin. The problem then reduces to finding the set $\Omega$ of fixed area $s$ that maximizes the integral of this radial, decreasing function. By the bathtub principle, this set is necessarily the centered disc $D_s$.

Key Contributions and Results

• Confirmation of the Conjecture: The paper proves that for any measurable set $\Omega$ with finite area $s$, the Ky Fan norm is maximized by the disc $D_s$ of area $s$. The sharp bound is given by:
  $$\sum_{j=1}^N \lambda_j(\Omega) \le \sum_{j=1}^N \lambda_j(D_s) = N - e^{-s} \sum_{k=0}^{N-1} \frac{(N-k)s^k}{k!}$$
  This result holds without assumptions of smoothness, boundedness, or connectedness on $\Omega$.
• Isoperimetric Schatten Inequalities: As a corollary, the paper establishes isoperimetric inequalities for all Schatten $p$-norms ($0 < p < \infty$). It shows that discs maximize the Schatten $p$-norms for $p > 1$ and minimize the quasi-norms for $0 < p < 1$ (under the condition that the operator is in the respective class). This extends previous results which were limited to radial sets or specific values of $p$ (e.g., $p=2$).
• Technical Novelty: The work introduces methods from quantum information theory (specifically Fock rearrangements of density operators) to solve a classical shape optimization problem in time-frequency analysis. This approach bypasses the differential inequality techniques used in previous Faber–Krahn proofs, which were insufficient for sums of eigenvalues.

Significance and Scope
The paper claims to resolve a significant open problem regarding the "accumulation of spectrograms," demonstrating that while individual higher-order eigenvalues may favor non-circular shapes (like annuli), the cumulative energy of the first $N$ modes is strictly optimized by the disc.

The author explicitly notes the limitations of the current work:

• The method does not solve the shape optimization problem for individual eigenvalues $\lambda_k(\Omega)$ where $k > 1$ (Problem P.1 in the introduction), which remains an open challenge due to symmetry breaking.
• The proof establishes the existence of the disc as a maximizer but does not address the uniqueness of the optimal set up to translations and null sets.
• The reliance on the specific structure of the Fock space and analyticity (via De Palma's theorem) means the results are currently specific to this setting, though the paper notes that restricted two-sided inequalities exist for more general windows.

In summary, the paper provides a definitive positive answer to the Nicola-Riccardi-Tilli conjecture for the Ky Fan norm, utilizing a "completely new perspective" derived from quantum information theory to handle shape optimization in the absence of radial symmetry.